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A218858
Number of Gaussian primes at taxicab distance n from the origin.
5
0, 0, 4, 12, 0, 16, 0, 20, 0, 16, 0, 28, 0, 24, 0, 32, 0, 32, 0, 36, 0, 24, 0, 36, 0, 64, 0, 32, 0, 48, 0, 44, 0, 32, 0, 72, 0, 64, 0, 48, 0, 72, 0, 60, 0, 56, 0, 60, 0, 40, 0, 56, 0, 72, 0, 112, 0, 64, 0, 76, 0, 88, 0, 56, 0, 136, 0, 92, 0, 80, 0, 76, 0, 88, 0
OFFSET
0,3
COMMENTS
Except for n = 2, there are no Gaussian primes at an even taxicab distance from the origin. All terms are multiples of 4. See A218859 for this sequence divided by 4.
The arithmetic derivative of Gaussian primes is either 1, -1, I, or -I.
LINKS
EXAMPLE
In the taxicab distance, the four Gaussian primes closest to the origin are 1+I, -1+I, -i-I, and 1-I. The 12 at taxicab distance 3 are the four reflections of 3, 2+I, and 1+2I.
MATHEMATICA
Table[cnt = 0; Do[If[PrimeQ[n - i + I*i, GaussianIntegers -> True], cnt++], {i, 0, n}]; Do[If[PrimeQ[i - n + I*i, GaussianIntegers -> True], cnt++], {i, n - 1, 0, -1}]; Do[If[PrimeQ[i - n - I*i, GaussianIntegers -> True], cnt++], {i, 1, n}]; Do[If[PrimeQ[n - i - I*i, GaussianIntegers -> True], cnt++], {i, n - 1, 1, -1}]; cnt, {n, 0, 100}]
CROSSREFS
Cf. A055025 (norms of Gaussian primes).
Cf. A222593 (first-quadrant Gaussian primes).
Cf. A225071, A225072 (number of terms at an odd distance from the origin).
Sequence in context: A222316 A255383 A354777 * A014458 A099733 A350259
KEYWORD
nonn
AUTHOR
T. D. Noe, Nov 12 2012
STATUS
approved